<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>

<link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; --title-bar-height:20px; }
.mac-os-11 { --title-bar-height:28px; }
html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }
body { margin: 0px; padding: 0px; height: auto; inset: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; tab-size: 4; }
iframe { margin: auto; }
a.url { word-break: break-all; }
a:active, a:hover { outline: 0px; }
.in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); }
#write { margin: 0px auto; height: auto; width: inherit; word-break: normal; overflow-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; }
#write.first-line-indent p { text-indent: 2em; }
#write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; }
#write.first-line-indent li { margin-left: 2em; }
.for-image #write { padding-left: 8px; padding-right: 8px; }
body.typora-export { padding-left: 30px; padding-right: 30px; }
.typora-export .footnote-line, .typora-export li, .typora-export p { white-space: pre-wrap; }
.typora-export .task-list-item input { pointer-events: none; }
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}
#write li > figure:last-child { margin-bottom: 0.5rem; }
#write ol, #write ul { position: relative; }
img { max-width: 100%; vertical-align: middle; image-orientation: from-image; }
button, input, select, textarea { color: inherit; font: inherit; }
input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; }
*, ::after, ::before { box-sizing: border-box; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; }
p { line-height: inherit; }
h1, h2, h3, h4, h5, h6 { break-after: avoid-page; break-inside: avoid; orphans: 4; }
p { orphans: 4; }
h1 { font-size: 2rem; }
h2 { font-size: 1.8rem; }
h3 { font-size: 1.6rem; }
h4 { font-size: 1.4rem; }
h5 { font-size: 1.2rem; }
h6 { font-size: 1rem; }
.md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; }
.hidden { display: none; }
.md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; }
a { cursor: pointer; }
sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-radius: 4px; cursor: pointer; }
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#write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; }
figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; }
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tr { break-inside: avoid; break-after: auto; }
thead { display: table-header-group; }
table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; }
table.md-table td { min-width: 32px; }
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#write pre.ty-contain-cm { white-space: normal; }
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.md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; }
.CodeMirror-line, twitterwidget { break-inside: avoid; }
.footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; }
.footnotes + .footnotes { margin-top: 0px; }
.md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; background: 0px 0px; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; }
li div { padding-top: 0px; }
blockquote { margin: 1rem 0px; }
li .mathjax-block, li p { margin: 0.5rem 0px; }
li blockquote { margin: 1rem 0px; }
li { margin: 0px; position: relative; }
blockquote > :last-child { margin-bottom: 0px; }
blockquote > :first-child, li > :first-child { margin-top: 0px; }
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}
.footnote-line { margin-top: 0.714em; font-size: 0.7em; }
a img, img a { cursor: pointer; }
pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgb(204, 204, 204); display: block; overflow-x: hidden; }
p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; }
#write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; }
p > .md-image:only-child { display: inline-block; width: 100%; }
#write .MathJax_Display { margin: 0.8em 0px 0px; }
.md-math-block { width: 100%; }
.md-math-block:not(:empty)::after { display: none; }
.MathJax_ref { fill: currentcolor; }
[contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; }
.md-task-list-item { position: relative; list-style-type: none; }
.task-list-item.md-task-list-item { padding-left: 0px; }
.md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; }
.math { font-size: 1rem; }
.md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px; }
.md-toc-content { position: relative; margin-left: 0px; }
.md-toc-content::after, .md-toc::after { display: none; }
.md-toc-item { display: block; color: rgb(65, 131, 196); }
.md-toc-item a { text-decoration: none; }
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.md-toc-h2 .md-toc-inner { margin-left: 2em; }
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.md-toc-h4 .md-toc-inner { margin-left: 6em; }
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  .md-toc-h6 .md-toc-inner { margin-left: 8em; }
}
a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; }
.footnote-line a:not(.reversefootnote) { color: inherit; }
.md-attr { display: none; }
.md-fn-count::after { content: "."; }
code, pre, samp, tt { font-family: var(--monospace); }
kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; }
.md-comment { color: rgb(162, 127, 3); opacity: 0.8; font-family: var(--monospace); }
code { text-align: left; vertical-align: initial; }
a.md-print-anchor { white-space: pre !important; border-width: initial !important; border-style: none !important; border-color: initial !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; background: 0px 0px !important; text-decoration: initial !important; text-shadow: initial !important; }
.os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; }
.md-diagram-panel > svg { max-width: 100%; }
[lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; }
[lang="mermaid"] .node text { font-size: 1rem; }
table tr th { border-bottom: 0px; }
video { max-width: 100%; display: block; margin: 0px auto; }
iframe { max-width: 100%; width: 100%; border: none; }
.highlight td, .highlight tr { border: 0px; }
mark { background: rgb(255, 255, 0); color: rgb(0, 0, 0); }
.md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; }
.md-expand mark .md-meta { opacity: 0.3 !important; }
mark .md-meta { color: rgb(0, 0, 0); }
@media print {
  .typora-export h1, .typora-export h2, .typora-export h3, .typora-export h4, .typora-export h5, .typora-export h6 { break-inside: avoid; }
}
.md-diagram-panel .messageText { stroke: none !important; }
.md-diagram-panel .start-state { fill: var(--node-fill); }
.md-diagram-panel .edgeLabel rect { opacity: 1 !important; }
.md-require-zoom-fix { height: auto; margin-top: 16px; margin-bottom: 16px; }
.md-require-zoom-fix foreignobject { font-size: var(--mermaid-font-zoom); }
.md-fences.md-fences-math { font-size: 1em; }
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.md-fences-advanced:not(.md-focus) { background: inherit; }
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.typora-export-show-outline #write { --webkit-flex:2; flex: 2 1 0%; }
.typora-export-sidebar .outline-content { position: fixed; top: 0px; max-height: 100%; overflow: hidden auto; padding-bottom: 30px; padding-top: 60px; width: 300px; }
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@media screen and (max-width: 800px) {
  .typora-export-sidebar { display: none; }
}
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.outline-content ul { margin-top: 0px; margin-bottom: 0px; }
.outline-content strong { font-weight: 400; }
.outline-expander { width: 1rem; height: 1.42857rem; position: relative; display: table-cell; vertical-align: middle; cursor: pointer; padding-left: 4px; }
.outline-expander::before { content: ""; position: relative; font-family: Ionicons; display: inline-block; font-size: 8px; vertical-align: middle; }
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:root {
    --side-bar-bg-color: #fafafa;
    --control-text-color: #777;
}

@include-when-export url(https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext);

/* open-sans-regular - latin-ext_latin */
  /* open-sans-italic - latin-ext_latin */
    /* open-sans-700 - latin-ext_latin */
    /* open-sans-700italic - latin-ext_latin */
  html {
    font-size: 16px;
    -webkit-font-smoothing: antialiased;
}

body {
    font-family: "Open Sans","Clear Sans", "Helvetica Neue", Helvetica, Arial, 'Segoe UI Emoji', sans-serif;
    color: rgb(51, 51, 51);
    line-height: 1.6;
}

#write {
    max-width: 860px;
  	margin: 0 auto;
  	padding: 30px;
    padding-bottom: 100px;
}

@media only screen and (min-width: 1400px) {
	#write {
		max-width: 1024px;
	}
}

@media only screen and (min-width: 1800px) {
	#write {
		max-width: 1200px;
	}
}

#write > ul:first-child,
#write > ol:first-child{
    margin-top: 30px;
}

a {
    color: #4183C4;
}
h1,
h2,
h3,
h4,
h5,
h6 {
    position: relative;
    margin-top: 1rem;
    margin-bottom: 1rem;
    font-weight: bold;
    line-height: 1.4;
    cursor: text;
}
h1:hover a.anchor,
h2:hover a.anchor,
h3:hover a.anchor,
h4:hover a.anchor,
h5:hover a.anchor,
h6:hover a.anchor {
    text-decoration: none;
}
h1 tt,
h1 code {
    font-size: inherit;
}
h2 tt,
h2 code {
    font-size: inherit;
}
h3 tt,
h3 code {
    font-size: inherit;
}
h4 tt,
h4 code {
    font-size: inherit;
}
h5 tt,
h5 code {
    font-size: inherit;
}
h6 tt,
h6 code {
    font-size: inherit;
}
h1 {
    font-size: 2.25em;
    line-height: 1.2;
    border-bottom: 1px solid #eee;
}
h2 {
    font-size: 1.75em;
    line-height: 1.225;
    border-bottom: 1px solid #eee;
}

/*@media print {
    .typora-export h1,
    .typora-export h2 {
        border-bottom: none;
        padding-bottom: initial;
    }

    .typora-export h1::after,
    .typora-export h2::after {
        content: "";
        display: block;
        height: 100px;
        margin-top: -96px;
        border-top: 1px solid #eee;
    }
}*/

h3 {
    font-size: 1.5em;
    line-height: 1.43;
}
h4 {
    font-size: 1.25em;
}
h5 {
    font-size: 1em;
}
h6 {
   font-size: 1em;
    color: #777;
}
p,
blockquote,
ul,
ol,
dl,
table{
    margin: 0.8em 0;
}
li>ol,
li>ul {
    margin: 0 0;
}
hr {
    height: 2px;
    padding: 0;
    margin: 16px 0;
    background-color: #e7e7e7;
    border: 0 none;
    overflow: hidden;
    box-sizing: content-box;
}

li p.first {
    display: inline-block;
}
ul,
ol {
    padding-left: 30px;
}
ul:first-child,
ol:first-child {
    margin-top: 0;
}
ul:last-child,
ol:last-child {
    margin-bottom: 0;
}
blockquote {
    border-left: 4px solid #dfe2e5;
    padding: 0 15px;
    color: #777777;
}
blockquote blockquote {
    padding-right: 0;
}
table {
    padding: 0;
    word-break: initial;
}
table tr {
    border: 1px solid #dfe2e5;
    margin: 0;
    padding: 0;
}
table tr:nth-child(2n),
thead {
    background-color: #f8f8f8;
}
table th {
    font-weight: bold;
    border: 1px solid #dfe2e5;
    border-bottom: 0;
    margin: 0;
    padding: 6px 13px;
}
table td {
    border: 1px solid #dfe2e5;
    margin: 0;
    padding: 6px 13px;
}
table th:first-child,
table td:first-child {
    margin-top: 0;
}
table th:last-child,
table td:last-child {
    margin-bottom: 0;
}

.CodeMirror-lines {
    padding-left: 4px;
}

.code-tooltip {
    box-shadow: 0 1px 1px 0 rgba(0,28,36,.3);
    border-top: 1px solid #eef2f2;
}

.md-fences,
code,
tt {
    border: 1px solid #e7eaed;
    background-color: #f8f8f8;
    border-radius: 3px;
    padding: 0;
    padding: 2px 4px 0px 4px;
    font-size: 0.9em;
}

code {
    background-color: #f3f4f4;
    padding: 0 2px 0 2px;
}

.md-fences {
    margin-bottom: 15px;
    margin-top: 15px;
    padding-top: 8px;
    padding-bottom: 6px;
}


.md-task-list-item > input {
  margin-left: -1.3em;
}

@media print {
    html {
        font-size: 13px;
    }
    table,
    pre {
        page-break-inside: avoid;
    }
    pre {
        word-wrap: break-word;
    }
}

.md-fences {
	background-color: #f8f8f8;
}
#write pre.md-meta-block {
	padding: 1rem;
    font-size: 85%;
    line-height: 1.45;
    background-color: #f7f7f7;
    border: 0;
    border-radius: 3px;
    color: #777777;
    margin-top: 0 !important;
}

.mathjax-block>.code-tooltip {
	bottom: .375rem;
}

.md-mathjax-midline {
    background: #fafafa;
}

#write>h3.md-focus:before{
	left: -1.5625rem;
	top: .375rem;
}
#write>h4.md-focus:before{
	left: -1.5625rem;
	top: .285714286rem;
}
#write>h5.md-focus:before{
	left: -1.5625rem;
	top: .285714286rem;
}
#write>h6.md-focus:before{
	left: -1.5625rem;
	top: .285714286rem;
}
.md-image>.md-meta {
    /*border: 1px solid #ddd;*/
    border-radius: 3px;
    padding: 2px 0px 0px 4px;
    font-size: 0.9em;
    color: inherit;
}

.md-tag {
    color: #a7a7a7;
    opacity: 1;
}

.md-toc { 
    margin-top:20px;
    padding-bottom:20px;
}

.sidebar-tabs {
    border-bottom: none;
}

#typora-quick-open {
    border: 1px solid #ddd;
    background-color: #f8f8f8;
}

#typora-quick-open-item {
    background-color: #FAFAFA;
    border-color: #FEFEFE #e5e5e5 #e5e5e5 #eee;
    border-style: solid;
    border-width: 1px;
}

/** focus mode */
.on-focus-mode blockquote {
    border-left-color: rgba(85, 85, 85, 0.12);
}

header, .context-menu, .megamenu-content, footer{
    font-family: "Segoe UI", "Arial", sans-serif;
}

.file-node-content:hover .file-node-icon,
.file-node-content:hover .file-node-open-state{
    visibility: visible;
}

.mac-seamless-mode #typora-sidebar {
    background-color: #fafafa;
    background-color: var(--side-bar-bg-color);
}

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}

/*.html-for-mac {
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#md-notification .btn {
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}

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}

.ty-preferences .window-content {
    background-color: #fafafa;
}

.ty-preferences .nav-group-item.active {
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    background: #999;
}

.menu-item-container a.menu-style-btn {
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    background-image: linear-gradient( 180deg , hsla(0, 0%, 100%, 0.8), hsla(0, 0%, 100%, 0)); 
}



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  direction: ltr;
}

mjx-container[jax="SVG"] > svg {
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  min-height: 1px;
  min-width: 1px;
}

mjx-container[jax="SVG"] > svg a {
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  stroke: blue;
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  left: 0px;
  clip: rect(1px, 1px, 1px, 1px);
  padding: 1px 0px 0px 0px !important;
  border: 0px !important;
  display: block !important;
  width: auto !important;
  overflow: hidden !important;
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  -webkit-user-select: none;
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g[data-mml-node="xypic"] path {
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</style><title></title>
</head>
<body class='typora-export os-windows'><div class='typora-export-content'>
<div id='write'  class=''><p><span>xgboost推导（涉及模型、损失函数，部分loss）：</span></p><p>&nbsp;</p><p><span>xgboost由树迭代生成，比如迭代次数为k-1：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n4" cid="n4" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="15.939ex" height="6.757ex" role="img" focusable="false" viewBox="0 -1740.7 7045.2 2986.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.819ex;"><defs><path id="MJX-1-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-1-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-1-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-1-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-1-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-1-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1-TEX-LO-2211" d="M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 1174 838T1033 881T882 898T699 902H574H543H251L259 891Q722 258 724 252Q725 250 724 246Q721 243 460 -56L196 -356Q196 -357 407 -357Q459 -357 548 -357T676 -358Q812 -358 896 -353T1063 -332T1204 -283T1307 -196Q1328 -170 1348 -124H1388Q1388 -125 1381 -145T1356 -210T1325 -294L1267 -449L666 -450Q64 -450 61 -448Q55 -446 55 -439Q55 -437 57 -433L590 177Q590 178 557 222T452 366T322 544L56 909L55 924Q55 945 60 948Z"></path><path id="MJX-1-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-1-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-1-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-1-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-1-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(300.6,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-1-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(490,0)"><use data-c="28" xlink:href="#MJX-1-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(879,0)"><use data-c="1D465" xlink:href="#MJX-1-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1451,0)"><use data-c="29" xlink:href="#MJX-1-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2117.8,0)"><use data-c="3D" xlink:href="#MJX-1-TEX-N-3D"></use></g><g data-mml-node="munderover" transform="translate(3173.6,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-1-TEX-LO-2211"></use></g><g data-mml-node="TeXAtom" transform="translate(148.2,-1087.9) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-1-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="3D" xlink:href="#MJX-1-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-1-TEX-N-31"></use></g></g><g data-mml-node="TeXAtom" transform="translate(537.8,1150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-1-TEX-I-1D458"></use></g></g></g><g data-mml-node="msub" transform="translate(4784.2,0)"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-1-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(617,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(5695.2,0)"><use data-c="28" xlink:href="#MJX-1-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(6084.2,0)"><use data-c="1D465" xlink:href="#MJX-1-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(6656.2,0)"><use data-c="29" xlink:href="#MJX-1-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></munderover><msub><mi>T</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>此时需要训练第k棵树，此时损失函数为：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block 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data-c="7B" xlink:href="#MJX-5-TEX-N-7B"></use></g><g data-mml-node="msup" transform="translate(5772,0)"><g data-mml-node="mi"><use data-c="1D454" xlink:href="#MJX-5-TEX-I-1D454"></use></g><g data-mml-node="TeXAtom" transform="translate(510,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-5-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D456" xlink:href="#MJX-5-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(734,0)"><use data-c="29" xlink:href="#MJX-5-TEX-N-29"></use></g></g></g><g data-mml-node="msub" transform="translate(7126,0)"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-5-TEX-I-1D463"></use></g><g data-mml-node="mi" transform="translate(518,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-5-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(8160.2,0)"><use data-c="2B" xlink:href="#MJX-5-TEX-N-2B"></use></g><g data-mml-node="mfrac" transform="translate(9160.4,0)"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="31" xlink:href="#MJX-5-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="32" xlink:href="#MJX-5-TEX-N-32"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(10100.4,0)"><use data-c="28" xlink:href="#MJX-5-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(10489.4,0)"><use data-c="1D6FD" xlink:href="#MJX-5-TEX-I-1D6FD"></use></g><g data-mml-node="mo" transform="translate(11277.7,0)"><use data-c="2B" xlink:href="#MJX-5-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(12277.9,0)"><g data-mml-node="mi"><use data-c="210E" xlink:href="#MJX-5-TEX-I-210E"></use></g><g data-mml-node="TeXAtom" transform="translate(609,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-5-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D456" xlink:href="#MJX-5-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(734,0)"><use data-c="29" xlink:href="#MJX-5-TEX-N-29"></use></g></g></g><g data-mml-node="mo" transform="translate(13731,0)"><use data-c="29" xlink:href="#MJX-5-TEX-N-29"></use></g><g data-mml-node="msubsup" transform="translate(14120,0)"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-5-TEX-I-1D463"></use></g><g data-mml-node="mn" transform="translate(518,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-5-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(518,-247) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-5-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(15041.5,0)"><use data-c="7D" xlink:href="#MJX-5-TEX-N-7D"></use></g><g data-mml-node="mo" transform="translate(15763.7,0)"><use data-c="2B" xlink:href="#MJX-5-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(16764,0)"><use data-c="1D706" xlink:href="#MJX-5-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(17347,0)"><use data-c="1D461" xlink:href="#MJX-5-TEX-I-1D461"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>L</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mi>i</mi><mi>t</mi></munderover><mo fence="false" stretchy="false">{</mo><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><msub><mi>v</mi><mi>i</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>β</mi><mo>+</mo><msup><mi>h</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><msubsup><mi>v</mi><mi>i</mi><mn>2</mn></msubsup><mo fence="false" stretchy="false">}</mo><mo>+</mo><mi>λ</mi><mi>t</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>对v求导取0得到，最佳的叶子节点取值：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n16" cid="n16" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="20.909ex" height="9.584ex" role="img" focusable="false" viewBox="0 -2368 9241.6 4236" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -4.226ex;"><defs><path id="MJX-6-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 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143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-6-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-6-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-6-TEX-I-1D6FD" d="M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 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46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-6-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,1424.7)"><g data-mml-node="mtd"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D454" xlink:href="#MJX-6-TEX-I-1D454"></use></g><g data-mml-node="TeXAtom" transform="translate(510,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-6-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D456" xlink:href="#MJX-6-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(734,0)"><use data-c="29" xlink:href="#MJX-6-TEX-N-29"></use></g></g></g><g data-mml-node="mo" transform="translate(1576.3,0)"><use data-c="2B" xlink:href="#MJX-6-TEX-N-2B"></use></g><g data-mml-node="mo" transform="translate(2576.5,0)"><use data-c="28" xlink:href="#MJX-6-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2965.5,0)"><use data-c="1D6FD" xlink:href="#MJX-6-TEX-I-1D6FD"></use></g><g data-mml-node="mo" transform="translate(3753.7,0)"><use data-c="2B" xlink:href="#MJX-6-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(4754,0)"><g data-mml-node="mi"><use data-c="210E" xlink:href="#MJX-6-TEX-I-210E"></use></g><g data-mml-node="TeXAtom" transform="translate(609,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-6-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D456" xlink:href="#MJX-6-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(734,0)"><use data-c="29" xlink:href="#MJX-6-TEX-N-29"></use></g></g></g><g data-mml-node="mo" transform="translate(6207.1,0)"><use data-c="29" xlink:href="#MJX-6-TEX-N-29"></use></g><g data-mml-node="msub" transform="translate(6596.1,0)"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-6-TEX-I-1D463"></use></g><g data-mml-node="mi" transform="translate(518,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-6-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(7685.8,0)"><use data-c="3D" xlink:href="#MJX-6-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(8741.6,0)"><use data-c="30" xlink:href="#MJX-6-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-894.7)"><g data-mml-node="mtd" transform="translate(1318.3,0)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-6-TEX-I-1D463"></use></g><g data-mml-node="mi" transform="translate(518,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-6-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(1089.7,0)"><use data-c="3D" xlink:href="#MJX-6-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2145.5,0)"><use data-c="2212" xlink:href="#MJX-6-TEX-N-2212"></use></g><g data-mml-node="mfrac" transform="translate(2923.5,0)"><g data-mml-node="msup" transform="translate(1163.7,676)"><g data-mml-node="mi"><use data-c="1D454" xlink:href="#MJX-6-TEX-I-1D454"></use></g><g data-mml-node="TeXAtom" transform="translate(510,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-6-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D456" xlink:href="#MJX-6-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(734,0)"><use data-c="29" xlink:href="#MJX-6-TEX-N-29"></use></g></g></g><g data-mml-node="mrow" transform="translate(220,-779.3)"><g data-mml-node="mi"><use data-c="1D6FD" xlink:href="#MJX-6-TEX-I-1D6FD"></use></g><g data-mml-node="mo" transform="translate(788.2,0)"><use data-c="2B" xlink:href="#MJX-6-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(1788.4,0)"><g data-mml-node="mi"><use data-c="210E" xlink:href="#MJX-6-TEX-I-210E"></use></g><g data-mml-node="TeXAtom" transform="translate(609,289) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-6-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D456" xlink:href="#MJX-6-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(734,0)"><use data-c="29" xlink:href="#MJX-6-TEX-N-29"></use></g></g></g></g><rect width="3441.5" height="60" x="120" y="220"></rect></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mi>β</mi><mo>+</mo><msup><mi>h</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><msub><mi>v</mi><mi>i</mi></msub><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>v</mi><mi>i</mi></msub><mo>=</mo><mo>−</mo><mfrac><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mrow><mi>β</mi><mo>+</mo><msup><mi>h</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfrac></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>将v代入原式：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n18" cid="n18" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" 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stretchy="false">)</mo></mrow></msup></mrow></mfrac><mo>+</mo><mi>λ</mi><mi>t</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p>&nbsp;</p><p><span>xgboost实现：</span></p><ul><li><p><span>bfs：breadth first search 一层一层构建</span></p></li><li><p><span>是否分裂：最大层数 和 增益</span></p></li><li><p><span>缺失值处理：枚举进入左节点更优，还是右节点更优</span></p></li><li><p><span>加速</span></p><ul><li><p><span>并行化（节点和特征）</span></p></li><li><p><span>当gready时，根据特征大小预排序，如 [样本3,    样本2,    样本0,   样本1]， 依次遍历，只需要     right.sum_g = parent.sum_g - 样本.g 就好了</span></p></li><li><p><span>cpu缓存加速，将需要的梯度信息，先放到cpu缓存里，不用到内存里读取</span></p><ul><li><p><span>第一层循环为feature，内层循环为node</span></p></li><li><p><span>根据feature大小，重新组织g，h，以及position信息</span></p></li><li><p><span>这些重新组织信息会用三次：</span></p><ul><li><span>遍历知道每个node上的最优分隔点</span></li><li><span>根据最优分隔点，更新position信息</span></li></ul></li></ul></li><li><p><span>使用h加权分桶，减少分裂点个数</span></p></li></ul></li></ul><p>&nbsp;</p><p>&nbsp;</p><p><span>lightGBM：</span></p><p><span>动机：解决训练数据太大，内存不够问题</span></p><ul><li><p><span>xgboost greedy模式，存储了排序index，消耗内存</span></p></li><li><p><span>xgboost 精确划分，消耗大</span></p></li><li><p><span>cache miss</span></p><ul><li><span>特征1：  样本idx -&gt; 样本g，h     其中样本g，h 存储在不同地方</span></li></ul></li></ul><p>&nbsp;</p><p><span>优化：</span></p><ul><li><p><span>直方图</span></p><ul><li><span>特征float 转为 箱子编号int，省内存</span></li><li><span>遍历k次</span></li><li><span>较粗的分隔点，有正则化效果</span></li><li><span>直方图做差加速</span></li></ul></li><li><p><span>带深度leaf-wise，不按层生长</span></p></li><li><p><span>gradient-based one side sampling 基于梯度的单边采样</span></p><ul><li><span>使用h作为样本权重</span></li></ul></li><li><p><span>exclusive feature bundle</span></p></li><li><p><span>类别特征支持</span></p><ul><li><p><span>one-hot缺点</span></p><ul><li><span>one vs rest</span></li><li><span>由于单类样本量少，增益小</span></li><li><span>小空间中，统计信息不准确</span></li></ul></li><li><p><span>取sumg/（sumh+beta）作为值，排序后，分别从左往右，从右往做遍历寻找切分点</span></p><ul><li><span>从左往右，缺省值被放到右树木</span></li></ul></li></ul></li><li><p><span>cache friendly：</span></p><ul><li><span>猜测，每个节点都维护了每个特征的histgram信息</span></li><li><span>分裂时，继续计算子节点的histgram</span></li></ul></li></ul><p>&nbsp;</p><p><span>catboost：</span></p><p><span>离散特征处理：</span></p><ul><li><span>若干次permutation</span></li><li><span>计算：</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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xlink:href="#MJX-142-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(9553.6,0)"><use data-c="1D44E" xlink:href="#MJX-142-TEX-I-1D44E"></use></g></g><rect width="11745.6" height="60" x="120" y="220"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>前</mtext><mtext>面</mtext><mtext>的</mtext><mtext>该</mtext><mtext>类</mtext><mtext>别</mtext><mtext>样</mtext><mtext>本</mtext><mtext>目</mtext><mtext>标</mtext><mtext>变</mtext><mtext>量</mtext><mtext>的</mtext><mtext>和</mtext><mo>+</mo><mi>a</mi><mo>∗</mo><mi>p</mi><mi>r</mi><mi>i</mi><mi>o</mi><mi>r</mi></mrow><mrow><mtext>前</mtext><mtext>面</mtext><mtext>样</mtext><mtext>本</mtext><mtext>中</mtext><mtext>该</mtext><mtext>类</mtext><mtext>别</mtext><mtext>个</mtext><mtext>数</mtext><mo>+</mo><mi>a</mi></mrow></mfrac></math></mjx-assistive-mml></mjx-container><script type="math/tex">\frac{前面的该类别样本目标变量的和 + a*prior}{前面样本中该类别个数 + a}</script></li></ul><p><span>无偏梯度估计</span></p><ul><li><span>每个样本，对应1个模型，模型使用除该样本外的其他样本训练得到</span></li><li><span>样本通过对应模型给出梯度，给其他模型做训练</span></li></ul><p><span>使用对称树：</span></p><ul><li><span>相当于加了正则</span></li><li><span>预测速度快</span></li></ul><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p></div></div>
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